Open Access
October 1996 Potential kernel for two-dimensional random walk
Yasunari Fukai, Kôhei Uchiyama
Ann. Probab. 24(4): 1979-1992 (October 1996). DOI: 10.1214/aop/1041903213


It is proved that the potential kernel of a recurrent, aperiodic random walk on the integer lattice $\mathbb{Z}^2$ admits an asymptotic expansion of the form $$(2 \pi \sqrt{|Q|})^{-1} \ln Q(x_2, -x_1) + \const + |x|^{-1} U_1 (\omega^x) + |x|^{-2} U_2 (\omega^x) + \dots ,$$ where $|Q|$ and $Q(\theta)$ are, respectively, the determinant and the quadratic form of the covariance matrix of the increment X of the random walk, $\omega^x = x/|x|$ and the $U_k (\omega)$ are smooth functions of $\omega, |\omega| = 1$, provided k that all the moments of X are finite. Explicit forms of $U_1$ and $U_2$ are given in terms of the moments of X.


Download Citation

Yasunari Fukai. Kôhei Uchiyama. "Potential kernel for two-dimensional random walk." Ann. Probab. 24 (4) 1979 - 1992, October 1996.


Published: October 1996
First available in Project Euclid: 6 January 2003

zbMATH: 0879.60068
MathSciNet: MR1415236
Digital Object Identifier: 10.1214/aop/1041903213

Primary: 31C20 , 60J15 , 60J45

Keywords: Laplace discrete operator , Oscillatory integral , potential kernel , Two-dimensional random walk

Rights: Copyright © 1996 Institute of Mathematical Statistics

Vol.24 • No. 4 • October 1996
Back to Top